Isoparametric submanifolds

✍ Scribed by Wolf Strübing

• Book ID: 104642918
• Publisher: Springer
• Year: 1986
• Tongue: English
• Weight: 951 KB
• Volume: 20
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00149586

✦ Synopsis

We consider isometric immersions F: M --* h4 between Riemannian manifolds whose higher mean curvature forms [in the sense of the author, Manuscripta Math. 49 (1984), 177-194] are all (covariantly) constant or, equivalently, whose shape operator with respect to each parallel unit normal vector field along any curve in M has constant eigenvalues: As in the hypersurface case we call such immersions isoparametric. Among other results it is proved: Isoparametric surfaces of constant curvature spaces are symmetric, i.e. have parallel second fundamental form. The same is true for isoparametric K/ihlerian hypersurfaces of complex space forms.